Preconditioners for ill conditioned (block) Toeplitz systems: facts a

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1 Preconditioners for ill conditioned (block) Toeplitz systems: facts and ideas Department of Informatics, Athens University of Economics and Business, Athens, Greece. Joint work with D. Noutsos

2 Problem We are interested in the fast and efficient solution of nm nm BTTB systems T n,m (f)x = b where f is nonnegative real-valued belonging to C 2π,2π defined in the fundamental domain Q = ( π,π] 2 and is a priori known. The entries of the coefficient matrix are given by t j,k = 1 4π 2 Q f(x, y)e i(jx+ky) dxdy, for j = 0, ±1,...,±(n 1) and k = 0,...,±(m 1).

3 Connection between f and T nm (f) The main connection between T nm (f) and the generating function is described by the the following result: Theorem If f C[ π,π] 2 and c < C the extreme values of f(x, y) on Q. Then every eigenvalue λ of the block Toeplitz matrix T satisfies the strict inequalities c < λ < C Moreover as m, n then λ min (T nm (f)) c and λ max (T nm (f)) C

4 Crucial relationship The basis for the construction of effective preconditioners is described by the following theorem Theorem Let f, g 0 C[ π,π] 2 (f and g not identically zero). Then for every m, n the matrix Tnm 1 (g)t nm (f) has eigenvalues in the open interval (r, R), where r = inf Q f g and R = sup Q f g

5 Negative result for τ algebra preconditioners Theorem (Noutsos,Serra,Vassalos, TCS (2004)) Let f be equivalent to p k (x, y) = (2 2 cos(x)) k + (2 2 cos(y)) k with k 2 and let β be a fixed positive number independent of n. Then for every sequence {P n } with P n τ, n = (n 1, n 2 ), and such that uniformly with respect to ˆn, we have λ max (P 1 n T n (f)) β (1) (a) the minimal eigenvalue of Pn 1 T n (f) tends to zero. (b) the number #{λ(n) σ(p 1 n T n(f)) : λ(n) N(n) 0} tends to infinity as N(ˆn) tends to infinity.

6 Negative result for τ algebra preconditioners Theorem (Noutsos,Serra,Vassalos, TCS (2004)) Let f be equivalent to p k (x, y) = (2 2 cos(x)) k + (2 2 cos(y)) k with k 2 and let α be a fixed positive number independent of n = (n 1, n 2 ). Then for every sequence {P n } with P n τ and such that λ min (P 1 n T n(f)) α (2) uniformly with respect to n, we have (a) the maximal eigenvalue of P 1 n T n (f) tends to. (b) the number #{λ(n) σ(p 1 n T n(f)) : λ(n) N(n) } tends to infinity as N(n) tends to infinity.

7 How to solve: 2D- ill conditioned problem Direct methods: Levinson type methods cost O(n 2 m 3 ) ops while superfast methods O(nm 3 log 2 n). Stability problems. Not optimal. PCG method with matrix algebra preconditioners. Cost O(k(ǫ)nm log nm), where k(ǫ) is the required number of iterations and depends from the condition number of T nm. PCG method where the preconditioner is band Toeplitz matrix. Under some assumptions, the cost is optimal (O(nm log nm)). Multigrid methods: Promising but in early stages. Cost O(nm log nm) ops. G. Fiorentino and S. Serra-Capizzano (1996), T. Huckle and J. Staudacher (2002),H.W. Sun, X.Q. Jin and Q.S. Chang (2004), A. Arico, M. Donatelli and S. Serra-Capizzano (2004).

8 More on Band Preconditioners S. Serra-Capizzano (BIT, (1994)) and M. Ng (LAA, (1997)) proposed as preconditioner the band BTTB matrix generated by the minimum trigonometric polynomial g which has the same roots with f. Let f = g h with h > 0. Then D. Noutsos, S. Serra Capizzano and P. Vassalos (Numer. Math. (2006)) proposed as preconditioners the band BTTB matrix generated by g ĥ where ĥ: 1 is the trigonometric polynomial arises from the Fourier approximation on h. 2 arises from the Lagrange interpolation of h at 2D Chebyshev points, or from the interpolation of h using the 2D Fejer kernel.

9 Preliminaries We assume that the nonnegative function f has isolated zeros (x 1, y 1 ),(x 2, y 2 ),...,(x k, y k ), on Q each one of multiplicities (2µ 1, 2ν 1 ),(2µ 2, 2ν 2 ),..., (2µ k, 2ν k ). Then, f can be written as where g = f = g w k [(2 2 cos(x x i )) µ i + (2 2 cos(y y i )) ν i ] i=1 and w, is strictly positive on Q.

10 New Proposal For the system T nm (f)x = b we define and propose as a preconditioner the product of matrices K nm (f) = A nm ( w)t nm (g)a nm ( w) = A nm (h)t nm (g)a nm (h) with h = w, A nm {τ, C, H}, where {τ, C, H} is the set of matrices belonging to Block τ, Block Circulant and Block Hartley algebra, respectively.

11 Construction of 2D algebras A v [n] Q n τ v [n] i = πi 2 n+1 n+1 ( C = 2πi n F n = 1 n H v [n] i v [n] i ( sin(jv [n] i ) ) n e ijv[n] i = 2πi n Re(F n ) + Im(F n ) ) i,j=1 The matrices C(h),τ(h), H(h) can be written as ( ) A nm (h) = Q nm Diag f(v [nm] ) Qnm H where v [nm] = v [n] v [m] and Q nm = Q n Qm

12 Obviously K nm (f) has all the properties that a preconditioner must satisfied, i.e, is symmetric, is positive definite, The cost for the solution of the arbitrary system K nm (f)x = b is of order O(nm log nm). O(nm log nm) for the inversion of A nm (h) by 2D FFT, and O(nm) for the inversion of block band Toeplitz matrix T n,m (g) which can be done by multigrid methods. So, the only condition that must be fulfill, in order to be a competitive preconditioner, is the spectrum of [ K A nm (f) ] 1 Tnm (f) being bounded from above and below.

13 Useful Definition Definition We say that a function h is a ((k 1, k 2 ),(x 0, y 0 ))-smooth function if l 1+l 2 x l 1 y l 2 h(x 0, y 0 ) = 0, l 1 < k 1, l 2 < k 2, and l 1 +l 2 < max{k 1, k 2 } and l 1+l 2 x l 1 y l 2 h(x 0, y 0 ) is bounded for l 1 = k 1, l 2 = 0 and l 2 = k 2, l 1 = 0, and l 1 + l 2 = max{k 1, k 2 }, l 1 < k 1, l 2 < k 2.

14 τ Case: Clustering Theorem Let f belongs to the Wiener class. Then for every ǫ the spectrum of [K τ nm(f)] 1 T nm (f) lies in [1 ǫ, 1 + ǫ], for n, m sufficient large, except of O(m + n) outliers. Thus, we have weak clustering around unity of the spectrum of the preconditioned matrix.

15 τ Case: Bounds Theorem Let f C 2π,2π even function on Q with roots (x 1, y 1 ),(x 2, y 2 ),...,(x k, y k ), each one of multiplicities (2µ 1, 2ν 1 ), (2µ 2, 2ν 2 ),..., (2µ k, 2ν k ), respectively. If g is the trigonometric polynomial of minimal degree that rises the roots of f and h is (µ i 1,ν i 1) smooth function at the roots (x i, y i ), i = 1(1)k, then the spectrum of the preconditioned matrix P τ = [K τ nm(f)] 1 T nm (f) is bounded from above as well as bellow: c < λ min (P τ ) < λ max (P τ ) < C with c, C constants independent of n, m

16 Circulant Case: Clustering Theorem Let f belongs to the Wiener class on Q. Then for every ǫ the spectrum of [ K C nm (f) ] 1 Tnm (f) lies in [1 ǫ, 1 + ǫ] for n, m sufficient large except O(m + n) outliers. Thus, we have a weak clustering of the spectrum of the preconditioned matrix around unity.

17 Circulant Case: Bound of Spectrum Theorem Let f C 2π,2π even function on Q with roots (x 1, y 1 ),(x 2, y 2 ),...,(x k, y k ), on Q each one of multiplicities (2µ 1, 2ν 1 ),(2µ 2, 2ν 2 ),..., (2µ k, 2ν k ) respectively. If g is the trigonometric polynomial of minimal degree that rises the roots of f and h is (µ i,ν i ) smooth function at the roots (x i, y i ), i = 1(1)k then the spectrum of the preconditioned matrix P C = [ K C nm(f) ] 1 Tnm (f) is bounded above as well as bellow: c < λ min (P C ) < λ max (P C ) < C with c, C constants independent of n, m

18 Case where h is not smooth enough Let us suppose that f has roots at (x i, y i ), i = 1(1)k. If h = f g is not as smooth as the previous Theorems demand, then the spectrum of the preconditioned matrix could be unbound. For that case we make a smoothing correction and instead of h we use ĥ defined as { h(x) (x, y) Q/ Ω ĥ = i h(x i, y i ) + α i g i (x, y) (x, y) Ω i where Ω i = {(x, y) : (x i, y i ) (x, y) < ǫ i } and α i is defined such that h(ǫ i,ǫ i ) = h(x i, y i ) + α i g(ǫ i,ǫ i )

19 Figure: Smoothing of h(x, y) = ( x + y +1)(x 4 +y 4 ), by ĥ(x, y). (2 2cos(x)) 2 +(2 2cos(y)) 2

20 Numerical Experiments We compare our proposal with the already known band preconditioners: B, which is generated by the trigonometric polynomial g that raises the roots of f K which is the product g ĥ with ĥ being the trigonometric polynomial arising from the 2D Fejer kernel on the positive part h of f P and F which arise: from the Lagrange interpolation of h at 2D Chebyshev points, and from the approximation of h using the 2D Fourier expansion, respectively. For all the tests the stopping criterion was r k 2 r 0 2 < 10 5, the starting vector the zero one and the righthand side vector of the system was (1 1 1) T

21 Table: f 1 (x, y) = (x 4 + y 2 )( x + y 3 + 1) n = m B K 4,4 F 4,4 P 4,4 τ C

22 Table: f 2 (x, y) = (x 2 + y 2 )(x 4 + y 4 +.1) n = m B K 6,6 F 6,6 P 6,6 τ C * * * 24 * * *: The number of iterations exceeds 100 -: The matrix is singular.

23 Application to Toeplitz plus band systems Let f C 2π,2π even function on Q with roots (x 1, y 1 ),(x 2, y 2 ),...,(x k, y k ), on Q each one of multiplicities (2µ 1, 2ν 1 ),(2µ 2, 2ν 2 ),..., (2µ k, 2ν k ) respectively. Consider the system (T nm (f) + B)x = b, where B is a symmetric, p.d block band matrix. We can choose as preconditioner the A nm ( w)(t nm (g) + B)A nm ( w) where {τ, C, H} is the set of matrices belonging to Block τ, Block Circulant and Block Hartley algebra, respectively, and g(x, y) the trigonometric polynomial having the same roots with the same multiplicities with f.

24 Table: f(x, y) = ( x y )( x + y + 2) n = m I B τ(f) * 98 9

25 τ preconditioners in ill condition case We define as P nm the matrix Q nm Diag ( ) f(v [nm] ) Qnm H where Q nm is the orthogonal matrix diagonalize the τ algebra and v [nm] ij = ( πi n+1, πj m+1 ). Then, the following theorem holds true: Theorem Let f C 2π,2π even function on Q with root at (0, 0) with multiplicity (q, r) with q, r 2. Then, the spectrum of PnmT 1 nm (f) is clustered around unity. Moreover, for every n, m it holds that c < σ(p 1 nmt nm (f)) < C with c, C > 0 independent of n, m

26 Table: f(x, y) = ( x + y )(x 2 + y 2 + 1) n = m #I λ min (I) λ max (P) λ min (P) #P

27 Thank you very much for your attention!